The Generalized Lipkin-Meshkov-Glick Model and the Modified Algebraic Bethe Ansatz

TL;DR

This paper introduces a generalized, exactly-solvable Lipkin-Meshkov-Glick model using a modified algebraic Bethe ansatz, expanding the class of integrable fermionic models related to Gaudin systems.

Contribution

It proposes a new integrable fermionic model generalizing the LMG model, solvable via a modified algebraic Bethe ansatz, linked to elliptic r-matrices and classical gyrostat quantization.

Findings

Explicit solutions for small fermion numbers N=1,2.

Connection of the model to the quantization of the Zhukovsky-Volterra gyrostat.

Extension of the LMG model within the Gaudin-type framework.

Abstract

We show that the Lipkin-Meshkov-Glick $2N$-fermion model is a particular case of one-spin Gaudin-type model in an external magnetic field corresponding to a limiting case of non-skew-symmetric elliptic $r$-matrix and to an external magnetic field directed along one axis. We propose an exactly-solvable generalization of the Lipkin-Meshkov-Glick fermion model based on the Gaudin-type model corresponding to the same $r$-matrix but arbitrary external magnetic field. This model coincides with the quantization of the classical Zhukovsky-Volterra gyrostat. We diagonalize the corresponding quantum Hamiltonian by means of the modified algebraic Bethe ansatz. We explicitly solve the corresponding Bethe-type equations for the case of small fermion number $N=1,2$.